CAIE P1 (Pure Mathematics 1) 2010 November

1 Find the term independent of \(x\) in the expansion of \(\left( x - \frac { 1 } { x ^ { 2 } } \right) ^ { 9 }\).

2 Points \(A , B\) and \(C\) have coordinates \(( 2,5 ) , ( 5 , - 1 )\) and \(( 8,6 )\) respectively.

1. Find the coordinates of the mid-point of \(A B\).
2. Find the equation of the line through \(C\) perpendicular to \(A B\). Give your answer in the form \(a x + b y + c = 0\).

3 Solve the equation \(15 \sin ^ { 2 } x = 13 + \cos x\) for \(0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\).

4

1. Sketch the curve \(y = 2 \sin x\) for \(0 \leqslant x \leqslant 2 \pi\).
2. By adding a suitable straight line to your sketch, determine the number of real roots of the equation $$2 \pi \sin x = \pi - x$$ State the equation of the straight line.

5 A curve has equation \(y = \frac { 1 } { x - 3 } + x\).

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
2. Find the coordinates of the maximum point \(A\) and the minimum point \(B\) on the curve.

6 A curve has equation \(y = \mathrm { f } ( x )\). It is given that \(\mathrm { f } ^ { \prime } ( x ) = 3 x ^ { 2 } + 2 x - 5\).

1. Find the set of values of \(x\) for which f is an increasing function.
2. Given that the curve passes through \(( 1,3 )\), find \(\mathrm { f } ( x )\).

7
\includegraphics[max width=\textwidth, alt={}, center]{32a57386-2696-4fda-a3cb-ca0c5c3be432-3_778_816_255_662} The diagram shows the function f defined for \(0 \leqslant x \leqslant 6\) by $$\begin{aligned} & x \mapsto \frac { 1 } { 2 } x ^ { 2 } \quad \text { for } 0 \leqslant x \leqslant 2 ,
& x \mapsto \frac { 1 } { 2 } x + 1 \text { for } 2 < x \leqslant 6 . \end{aligned}$$

1. State the range of f .
2. Copy the diagram and on your copy sketch the graph of \(y = \mathrm { f } ^ { - 1 } ( x )\).
3. Obtain expressions to define \(\mathrm { f } ^ { - 1 } ( x )\), giving the set of values of \(x\) for which each expression is valid.

8
\includegraphics[max width=\textwidth, alt={}, center]{32a57386-2696-4fda-a3cb-ca0c5c3be432-3_600_883_1731_630} The diagram shows a rhombus \(A B C D\). Points \(P\) and \(Q\) lie on the diagonal \(A C\) such that \(B P D\) is an arc of a circle with centre \(C\) and \(B Q D\) is an arc of a circle with centre \(A\). Each side of the rhombus has length 5 cm and angle \(B A D = 1.2\) radians.

1. Find the area of the shaded region \(B P D Q\).
2. Find the length of \(P Q\).

9

1. A geometric progression has first term 100 and sum to infinity 2000. Find the second term.
2. An arithmetic progression has third term 90 and fifth term 80 .
   1. Find the first term and the common difference.
   2. Find the value of \(m\) given that the sum of the first \(m\) terms is equal to the sum of the first ( \(m + 1\) ) terms.
   3. Find the value of \(n\) given that the sum of the first \(n\) terms is zero.

10
\includegraphics[max width=\textwidth, alt={}, center]{32a57386-2696-4fda-a3cb-ca0c5c3be432-4_561_599_744_774} The diagram shows triangle \(O A B\), in which the position vectors of \(A\) and \(B\) with respect to \(O\) are given by $$\overrightarrow { O A } = 2 \mathbf { i } + \mathbf { j } - 3 \mathbf { k } \quad \text { and } \quad \overrightarrow { O B } = - 3 \mathbf { i } + 2 \mathbf { j } - 4 \mathbf { k } .$$ \(C\) is a point on \(O A\) such that \(\overrightarrow { O C } = p \overrightarrow { O A }\), where \(p\) is a constant.

1. Find angle \(A O B\).
2. Find \(\overrightarrow { B C }\) in terms of \(p\) and vectors \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\).
3. Find the value of \(p\) given that \(B C\) is perpendicular to \(O A\).

11
\includegraphics[max width=\textwidth, alt={}, center]{32a57386-2696-4fda-a3cb-ca0c5c3be432-5_710_931_255_607} The diagram shows parts of the curves \(y = 9 - x ^ { 3 }\) and \(y = \frac { 8 } { x ^ { 3 } }\) and their points of intersection \(P\) and \(Q\). The \(x\)-coordinates of \(P\) and \(Q\) are \(a\) and \(b\) respectively.

1. Show that \(x = a\) and \(x = b\) are roots of the equation \(x ^ { 6 } - 9 x ^ { 3 } + 8 = 0\). Solve this equation and hence state the value of \(a\) and the value of \(b\).
2. Find the area of the shaded region between the two curves.
3. The tangents to the two curves at \(x = c\) (where \(a < c < b\) ) are parallel to each other. Find the value of \(c\).